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I have been given a question which is very similar to this:

Apollonius circle, its radius and center

However, I have been told to translate and scale the given circle to give a unit circle centred at the origin, and then to show that for any point $a \in\Bbb{C}$ there is a unique $b \in \Bbb{C}$ and $k \in (0,1)$ such that $|z-a| = k|z-b|$.

Is there any way to do this without completing the square?

My lecturer told us to get it to the unit circle centred at the origin where he took $a = 1/2$, and we need to tackle it like this - but am I allowed to take $a =1/2$ or is this the generic case?

Update: I've managed to get a = 1/b and b = 1/a, but still cannot find k

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  • $\begingroup$ For some reason, my blackboard bold or 'in' signs won't work $\endgroup$ – ekudamram Oct 27 '13 at 15:53

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